Cayley’s parameterization of orthogonal matrices

Any orthogonal matrixMathworldPlanetmath O which does not have -1 as an eigenvalueMathworldPlanetmathPlanetmathPlanetmathPlanetmath can be expressed as


for some suitable skew-symmetric matrix A. Conversely, any skew-symmetric matrix A can be expressed in terms of a suitable orthogonal matrix O by a similarPlanetmathPlanetmath formulaMathworldPlanetmathPlanetmath,


These two formulae are each other’s inversesMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath and set up a one-to-one correspondence between orthogonalMathworldPlanetmathPlanetmath and skew-symmetric matrices.

0.0.1 Proof

The restrictionPlanetmathPlanetmathPlanetmath on the eigenvalues of O is necessary in order for I+O to be invertible.

It is a matter of simple computation why these formulae are correct. Suppose that A is skew-symmetric. Then


Using the fact that the transposeMathworldPlanetmath of a productPlanetmathPlanetmath is the product of the transposes in the opposite order,


Using the fact that the transpose of a sum is the sum of transposes and the transpose of an inverse is the inverse of the transpose,


By the definition of skew-symmetry, AT=-A and IT=I,


Finally, since I+A and I-A commute, we may switch the order of the second and third factors:


Then the first two factors and the last two factors cancel, showing that OTO=I.

Next, we verify that the second formula is indeed the inverse of the first formula. Multiplying by I-A on both sides,


Expanding this and moving terms from one side of the equation to the other,




Multiplying both sides by (I+O)-1, we obtain the desired formula:


Finally, one can show that, if O is orthogonal, then A is skew-symmetric using the same sort of computation that was used to show the converse:


Using the facts about transposes of sums, products, and inverses,


Since O is orthogonal, OT=O-1. As usual IT=I.


Insert an identity matrixMathworldPlanetmath between the two factors like so:


Replace the identity matrix with OO-1:


AbsorbingPlanetmathPlanetmath the O and the O-1 into the factors,




(O-I) and (O+I)-1 commute, and consequently


0.0.2 Cayley Transform

The relationMathworldPlanetmathPlanetmath between A and O which is set up by the formulas




is sometimes known as the Cayley transform. Note that the proof that these two formulas are each other’s inverses did not require A to be skew-symmetric or O to be orthogonal. Hence, the Cayley transform is defined for all matrices such that -1 is not an eigenvalue of O. (Recall that this condition is necessary to insure that O+I is invertible.

0.0.3 Generalizations

The Cayley parameterization can be generalized to unitaryMathworldPlanetmathPlanetmath transforms. Namely, if U is a unitary matrix, then U is the Cayley transform of a skew-Hermitean matrix A. Since a skew-Hermitean matrix can be written as i times a Hermitean matrix, the Cayley transform is often written as follows when dealing with unitary matrices:


where H is Hermitean. The proof in this case is substantially the same as was presented above; all one has to do is replace matrix transpositionMathworldPlanetmath with Hermitean conjugationMathworldPlanetmath.

A special case of this worth pointing out is the case of one-dimensional unitary matrices. The sole entry of a one dimensional unitary matrix must have modulus 1 and the sole entry of a one-dimensional Hermitean matrix must be real. In that case, the Cayley transform reduces to


which is a fractional linear transform that maps the unit circle to the real axis.

The Cayley parameterization can be generalized to the case of a general inner productMathworldPlanetmath with arbitrary signaturePlanetmathPlanetmathPlanetmathPlanetmath (see Sylvester’s law for the definition of signature — Cayley and Sylvester were the best of friends). We simply need to define the transpose of a matrix M by the condition (MTu)v=u(Mv) for all vectors u and v. In particular, this allows one to parameterize pseudo-orthogonal matrices such as Lorentz transformations using a Cayley parameterization. Likewise, given a conjugate linear inner product on a complex vector space, one has a Cayley parameterization of the unitary (or pseudo-unitary) transforms which preserve the product.

In conclusionMathworldPlanetmath, it might be worth pointing out that the Cayley transform generalizes to the case of infiniteMathworldPlanetmath dimensionsPlanetmathPlanetmathPlanetmath, if one replaces matrices with operatorsMathworldPlanetmath on a Hilbert spaceMathworldPlanetmath. In particular, it is useful because unitary and orthogonal operators are bounded whereas Hermitean and skew-symmetric operators may or may not be bounded. For instance, it is often easier to obtain the spectral decomposition of a Hermitean operator or study symmetricPlanetmathPlanetmathPlanetmathPlanetmath extensionsPlanetmathPlanetmathPlanetmath of a symmetric operator by first performing a Cayley transform and dealing with the resulting bounded operatorMathworldPlanetmathPlanetmath.

Title Cayley’s parameterization of orthogonal matrices
Canonical name CayleysParameterizationOfOrthogonalMatrices
Date of creation 2013-03-22 14:51:38
Last modified on 2013-03-22 14:51:38
Owner rspuzio (6075)
Last modified by rspuzio (6075)
Numerical id 20
Author rspuzio (6075)
Entry type Theorem
Classification msc 22E70
Classification msc 15A57
Defines Cayley transform